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1 Copyright 2008 by ASME

Proceedings of IMECE082008 ASME International Mechanical Engineering Congress and Exposition

October 31 - November 6, 2008, Boston, Massachusetts USA

IMECE2008-66017

DESIGN AND MANUFACTURE OF SPIRAL BEVEL GEARS WITH REDUCED TRANSMISSIONERRORS

V. SimonBudapest University of Technology and Economics

Faculty of Mechanical Engineering

Department of Machine and Product DesignH-1111 Budapest, Megyetem rkp. 3, Hungary

[email protected]

ABSTRACT

A method for the determination of the optimal polynomialfunctions for the conduction of machine-tool setting variationsin pinion teeth finishing in order to reduce the transmissionerrors in spiral bevel gears is presented. Polynomial functionsof order up to five are applied to conduct the variation of thecradle radial setting and of the cutting ratio in the process for

pinion teeth generation. Two cases were investigated: in the

first case the coefficients of the polynomial functions are con-stant throughout the whole generation process of one piniontooth-surface, in the second case the coefficients are differentfor the generation of the pinion tooth-surface on the two sidesof the initial contact point. The obtained results have shownthat by the use of two different fifth-order polynomial functionsfor the variation of the cradle radial setting for the generation ofthe pinion tooth-surface on the two sides of the initial contact

point, the maximum transmission error can be reduced by 81%.By the use of the optimal modified roll, this reduction is 61%.The obtained results have also shown that by the optimal varia-tion of the cradle radial setting, the influence of misalignmentsinherent in the spiral bevel gear pair and of the transmittedtorque on the increase of transmission errors can be considera-

bly reduced.

INTRODUCTION

The traditional cradle-type hypoid generators are evolvedinto computer numerical control (CNC) hypoid generatingmachines, as are the Gleason Phoenix series and theKlingelnberg universal spiral bevel gear generating machine.These new CNC hypoid generators have made it possible to

perform nonlinear correction motions for the cutting of theface-milled spiral bevel and hypoid gears. The following refer-ences summarize the proposed free-form cutting methods.

Local synthesis of spiral bevel gears with localized bearingcontact and predesigned parabolic function of a controlled levelfor transmission errors was proposed by Litvin and Zhang [1].The theory of modified roll, the variation of cutting ratio in the

process for pinion teeth generation was introduced. In the paperpublished by Argyris et al. [2], a computerized method of localsynthesis and simulation of meshing of spiral bevel gears with

pinion tooth-surface generated by applying a third-order func-tion for modified roll was presented. Fuentes et al. [3] and Lit-

vin and Fuentes [4] have developed an integrated computerizedapproach for the design and stress analysis of low-noise spiral

bevel gear drives with adjusted bearing contact. The predes-igned parabolic function of transmission errors was achieved bythe application of modified roll for pinion tooth generation.Ref. [5] covered the design, manufacturing, stress analysis andresults of experimental tests of prototypes of spiral bevel gearswith low levels of noise and vibration and increased endurance.Three shapes of blade profile and a modified roll for piniontooth generation were applied.

Based on the grinding mechanism and machine-tool settingsfor the Gleason modified roll hypoid grinder, a mathematicalmodel for the tooth geometry of spiral bevel and hypoid gearswas developed by Lin and Tsay [6].Chang et al. [7] proposed ageneral gear mathematical model simulating the generation

process of a 6-axis CNC hobbing machine. The so-called Uni-versal Motion Concept (UMC) was developed by Stadtfeld [8].In the UMC eight correction mechanisms were introduced intothe calculation of machine settings for gear cutting. In order todevelop a gear geometry that reduces gear noise and increasesthe strength of gears, the Universal Motion Concept was ap-

plied for hypoid gear design by Stadtfeld and Gaiser [9]. Higherorder kinematic freedoms up to the 4th order were applied toachieve the best possible result in noise, sensitivity, and adjust-ability.

Linke et al. [10] presented a method for taking any addi-tional motions mapped in the process-independent mathemati-

Proceedings of IMECE20082008 ASME International Mechanical Engineering Congress and Exposition

October 31-November 6, 2008, Boston, Massachusetts , USA

IMECE2008-66017
mailto:[email protected]:[email protected]:[email protected]

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cal model of the generating process into account in the simula-tion of the manufacturing process of bevel gears. It was demon-strated, how a specific influencing of the meshing and stressconditions can be achieved by such additional motions. Amathematical model of universal hypoid generator with sup-

plemental kinematic flank correction motions was proposed byFong [11] to simulate virtually all primary spiral bevel and

hypoid cutting methods. The supplemental kinematic flankcorrection motions, such as modified generating roll ratio, heli-cal motion, and cutter tilt were included into the proposedmathematical model. Wang and Fong in Ref. [12] proposed amethodology to improve the adjustability of the spiral bevelgear assembly by modifying the radial motion of the head cut-ter in the machine plane of the hypoid generator. A method tosynthesize the mating tooth-surfaces of a face-milled spiral

bevel gear set transmitting rotation with a predeterminedfourth-order motion curve and contact path was presented byWang and Fong [13].

By Achtmann and Br [14], modified helical motion andmodified roll were applied to produce optimally fitted bearingellipses. The paper published by Fan [15], presented the theoryof the Gleason face hobbing process. In Ref. [16], a genericmodel of tooth-surface generation for spiral bevel and hypoidgears produced by face-milling and face-hobbing processesconducted on free-form computer numerical control (CNC)hypoid gear generators was presented. Fan et al. [17] describeda new method of tooth flank form error correction, utilizing theuniversal motions and the universal generation model for spiral

bevel and hypoid gears. Shih and Fong [18] proposed a flankmodification methodology for face-hobbing spiral bevel gearand hypoid gears, based on the ease-off topography of the geardrive. Gear generation with supplemental spatial motions (heli-cal motion, tilt motion), particularly interesting for gear genera-tion with modern free-form cutting machines, was presented by

Di Puccio et al [19]. The application of modified roll and basicmachine root angle variation in spiral bevel pinion finishingwas introduced. In Ref. [20] published by Cao et al., third orderfunction for the modified roll was applied in order to achieve a

predesigned parabolic function of transmission errors and toimprove the contact pattern with the desired shape of contact

path in spiral bevel gears. Liu and Wang [21] presented amethod for realizing and improving the conventional gear cut-ting, associated with a traditional machine-tool upon a CNCfree-form gear cutting machine.

The truly conjugated spiral bevel gears are theoreticallywith line contact. In order to decrease the sensitivity of the gear

pair to errors in tooth-surfaces and to the mutual position of themating members, carefully chosen modifications are usually

introduced into the teeth of one or both members. As a result ofthese modifications, the spiral bevel gear pair becomes mis-matched, and a point contact of the meshed teeth-surfacesappears instead of line contact. In practice, these modificationsare usually introduced by applying the appropriate machine-tool setting for pinion and gear manufacture. The generation oftooth-surfaces of the pinion and the gear in mismatched spiral

bevel gears was described in Ref. [22]. In this paper a methodis presented for the determination of the optimal polynomialfunction for the conduction of machine-tool setting variation in

pinion teeth finishing in order to reduce the transmission errorsin mismatched spiral bevel gears.

NOMENCLATURE

c = sliding base setting for pinion finishing, mm

ne = composite manufacture and alignment error, mm

pe = basic radial for pinion finishing, mm

f = machine center to back, mm

g = blank offset for pinion finishing, mm

gpi = velocity ratio in the kinematic scheme of the machine-

tool for the generation of pinion tooth-surface

21 N,N = numbers of pinion and gear teeth

p = distance of the initial contact point from pinion apex,mm

maxp = maximum tooth contact pressure, Pa

1Tr = pinion finishing cutter radius, mm

s = geometrical separation of tooth-surfaces, mmT = transmitted torque, mN

1 = pinion finishing blade angle, deg.a = pinion offset, mm

b = displacement of the pinion along its axis, mmc = displacement of the pinion along the gear axis, mmF = concentrated load, N

ny = composite displacement of contacting surfaces, mm

2 = angular displacement of the driven gear, deg.

h = horizontal angular misalignment of pinion axis, deg.

v = vertical angular misalignment of pinion axis, deg.

21, = rotational angles of the pinion and the gear, deg.

2010 , = initial rotational angles of the pinion and the gear,deg.

1 = machine root angle for pinion finishing, deg.( )c = angular cradle velocity, 1s

1 = angle of pinion rotation during its generation, deg.

cp = cradle angle rotation for pinion finishing, deg.

0cp = initial cradle angle setting for pinion finishing, deg.

THEORETICAL BACKGROUND

The Spiral Bevel Gear Pair. A Gleason type spiral bevelgear pair with generated pinion and gear teeth is treated (seeFig. 1). The pinion is the driving member. The convex side ofthe gear tooth and the mating concave side of the pinion toothare the drive sides. The modifications are introduced into the

pinion tooth-surface by applying machine-tool setting varia-tions in pinion tooth generation. As a result of these modifica-tions the spiral bevel gear pair becomes mismatched and a pointcontact of the meshed teeth-surfaces appears instead of linecontact.

The relative position of the pinion and the gear is defined bythe following equation (based on Fig. 1):

01

vhvhv

hh

vhvhv

011202

1000

acoscossinsinsin

c0sincos

bpsincoscossincos

rrMrrrr

==

(1)

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',

2x

'

'

z 01z1

'z01

(2)

'

O01 , ,'

z02 z 2z 01z 02

zz 2z1

x1

01x

01x

02x

01x 01x 1x

01y

01y ' , 1y

'

02

01

h

'

p

2

v

1

02y 2y

02y

2y

01y 1y

01y

(1)

(1)

2

1

(2)

h

vO01 O1 O01

O2 O02O01

O1

O02

O01

O02 O2,O01 O01

b

a

x x,02 2

,

,

01

c

O2

Fig. 1:Relative position of the pinion and the gear in mesh

where matrix 12M defines the relation between the stationary

coordinate systems 01K and 02K , in which the pinion and the

gear are rotating through mesh. This matrix includes the possi-ble misalignments of the pinion: the pinion offset ( a ), thedisplacements of the pinion along the pinion axis ( b ) andalong the gear axis ( c ), and the angular misalignments of the

pinion axis in the horizontal ( h ) and in the vertical ( v ) plane.

Machine-Tool Setting for Pinion Finishing. The machine-tool setting used for pinion teeth finishing is given in Fig. 2.

The concave side of pinion teeth is in the coordinate system

1K (attached to the pinion, Fig. 2) defined by the following

system of equations

)T(

Tp1p2p3

)1(

11

1rMMMrrr

= (2a)

0)T(

1m

)1,T(

1m11 = ev

rr

(2b)

where( )11

T

Trr

is the radius vector of tool-surface points, matrices

p1M , p2M , and p3M provide the coordinate transformations

from system 1TK (rigidly connected to the cradle and head-

cutter 1T ) to system 1K (rigidly connected to the being gener-

ated pinion). Equation (2b) describes mathematically the gen-

eration of pinion tooth-surface by the head-cutter [22]. The

matrices and vectors of system of Eqs. (2a) and (2b) are definedas it follows.

The surface of the head-cutter used for finishing the con-

cave side of pinion teeth is in the coordinate system 1TK (at-

tached to the tool) defined by the following equation (based onFig. 2):

+

+

=

1

sin)tgur(

cos)tgur(

u

),u(1T

1T)T(

T

1

11

1rr

(3)

On the basis of Fig. 2 and Eq. (3), for the relative velocityvector

)1,T(

1m1v

r

of tool 1T to the pinion, and for the unit normal

vector of the tool-surface)T(

1m1e

r

, it follows

( )( )( )[ ]

+

+

=)T(

1m1

1T(

1m1

)T(

1mgp

1

)T(

1mgp

)T(

1m

1

)T(

1mgp

)c()1,T(

1m

111

11

1

1

ycoscxsinyi

singziz

cosgzi

vr

(4)

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y10

ym1

1

1

x

xm1

xT1

01

f1

k1

O1

1

OT1

m1y y

10

yT1

zT1

rt

m1z

M M

(c)

bf

z10

g

cp

T1O

m1O

10O

10O

1

O

r T1

10O

c

(1)

z10

z1

x10

x10

x1

1

u

f

1

T1z

e

p

m1O

Cutter T1

p

Fig. 2:Machine-tool setting for pinion tooth-surface finishing

==

0

sincos

coscos

sin

1

1

1

p1

)T(

1Tp1

)T(

1m

11 MeMerr

(5)

where( ) ( )1

1

1 T

Tp1

T

1m rMrrr

=

For matrices p1M , p2M , and p3M , providing the coordi-

nate transformations, on the basis of Fig. 2, it follows

1T

cppcpcp

cppcpcp

1Tp11m

1000

sinesincos0

cosecossin0

0001

rrMrrrr

==

(6)

1m

111

111

1mp201

1000

g100

sincf0cossin

cosc0sincos

rrMrrrr

== (7)

0111

11

01p31

1000

0cos0sin

p010

0sin0cos

rrMrrrr

== (8)

while, for the traditional cradle-type hypoid generators

( )0cpcpgp101 i += . Angles 10 and 0cp correspondto the generation of the initial contact point on the piniontooth-surface. Because of the mismatch of the gear pair, only inone point of the path of contact, called as the initial contact

point, the basic mating equation of the contacting pinion andgear tooth-surfaces is satisfied, producing the correct velocityratio based on the numbers of teeth. Usually, the middle pointof the gear tooth flank is chosen for the initial contact point andthe machine-tool settings for pinion and gear manufacture arecalculated due to this initial contact point.

In this paper the variations of the cradle radial setting pe

and of the modified roll for pinion generation are conducted bypolynomial functions of order up to five:

( ) ( ) ( ) ( ) ( )50cpcp5e4

0cpcp4e

3

0cpcp3e

2

0cpcp2e0cpcp1e0pp cccccee +++++= (9)

( ) ( ) ( ) ( ) ( )50cpcp4i4

0cpcp3i

3

0cpcp2i

2

0cpcp1i0cpcpgp1 cccci ++++= (10)

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where 1 is the angle of pinion rotation during its generation;

angle cp is the angle of rotation of the cradle of the cutting

machine.Two solutions are investigated: in the first case the coeffi-

cients in Eqs. (9) and (10) are constant throughout the wholegeneration process of one pinion tooth-surface, in the second

case the coefficients are different for the generation of the pin-ion tooth-surface on the two sides of the initial contact point.The results have shown that the second solution gives much

better results.Potential Contact Lines Under Load. As it was mentioned

before, as a result of the modifications introduced into the teethof one or both members, theoretically point contact of themeshed tooth-surfaces appears instead of the line contact.However, as the tooth-surface modifications are relativelysmall, even in the case of light loading, the theoretical pointcontact spreads over a narrow surface along the whole or partof the potential contact line. These potential contact lines can

be obtained by determining the line of minimal separations ofthe mating tooth-surfaces along the face width of teeth. The

separations are defined as the distances of the correspondingsurface points that are the intersection points of the straight line

parallel to the common surface normal in the instantaneouscontact point, with the pinion and gear tooth-surfaces. Mathe-matically, it means the minimization of the function

( ) ( )( ) ( ) ( )( ) ( ) ( )( )210220221

02

2

02

21

02

2

02 zzyyxxs ++= (11)

where( )102rr

and( )202rr

are the position vectors of the correspond-

ing points on the pinion and gear tooth-surfaces.It should be mentioned that in the case of edge contact the

computer program developed and applied automatically adjusts

the values of separations in other points of the potential contactline accordingly to the zero separation of the edge contact

point.The method for the determination of minimal separations

and potential contact lines is fully described in Ref. [22].Transmission Errors. The total transmission error consists

of the kinematical transmission error due to the mismatch of thegear pair and eventual tooth errors and misalignments of themeshing members, and of the transmission error caused by thedeflections of teeth.

It is assumed that the pinion is the driving member and thatis rotating at a constant velocity. As the result of the mismatchof gears, a varying angular velocity ratio of the gear pair and anangular displacement of the driven gear member from the theo-

retically exact position based on the ratio of the numbers ofteeth occurs. This angular displacement of the gear can be ex-

pressed as

( ) ( ) s221011202k

2 N/N += (12)

where 10and 20are the initial angular positions of the pinionand the gear corresponding to the initial contact point, 2is theinstantaneous angular position of the gear for a particular angu-

lar position of the pinion, 1 [22]; N1and N2are the numbers ofpinion and gear teeth, respectively, and s2 is the angulardisplacement of the gear due to edge contact in the case of

misalignments of the mating members when a negative sepa-ration occurs on a tooth pair different from the tooth pair forwhich the instantaneous angular position is calculated [22].

The angular displacement of the gear,( )d2 , caused by the

variation of the compliance of contacting pinion and gear teethrolling through mesh, will be determined in the load distribu-

tion calculation.Therefore, the total angular position error of the gear is de-

fined by the equation

( ) ( )d2

k

22 += (13)

Load Distribution. The load distribution calculation isbased on the conditions that the total angular position errors ofthe gear teeth being instantaneously in contact under load must

be the same, and along the contact line (contact area) of eachtooth pair instantaneously in contact, the composite displace-ments of tooth-surface points - as the sums of tooth deforma-tions, tooth surface separations, misalignments, and composite

tooth errors - should correspond to the angular position of thegear member. Therefore, in all the points of the instantaneouscontact lines, the following displacement compatibility equa-tion should be satisfied:

( ) ( ) ( ) ( )k2

0

D

nk

2

d

22r

y+

=+=

r

earr

rrr

(14)

where ny is the composite displacement of contacting sur-

faces in the direction of the unit tooth surface normal er

, rr

is

the position vector of the contact point, Dr is the distance of the

contact point to the gear axis, and 0ar

is the unit vector of the

gear axis.The composite displacement of the contacting surfaces in

contact point D, in the direction of the tooth-surface normal,can be expressed as

( ) ( ) ( )DnDDn zezszwy ++= (15)

where Dz is the coordinate of Point D along the contact line,

( )Dzw is the total deflection in Point D, ( )Dzs is the relativegeometrical separation of teeth-surfaces in Point D, and ( )Dn zeis the composite error in Point D, which is the sum of manufac-turing and alignment errors of pinion and gear.

The total deflection in Point D is defined by the followingequation: ([23, 24])

( ) ( ) ( ) ( ) ( ) +=itL

DDcFFDdD zpzKdzzpz,zKzw (16)

where itL is the geometrical length of the line of contact on

tooth pair ti , ( )FDd z,zK is the influence factor of tooth loadacting in tooth-surface Point F on total composite deflection of

pinion and gear teeth in contact Point D. dK includes the bend-

ing and shearing deflections of pinion and gear teeth, pinionand gear body bending and torsion, and deflections of support-

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ing shafts. A finite element computer program is developed forthe calculation of bending and shearing deflections in the pin-

ion and in the gear [25]. ( )Dc zK is the influence factor for thecontact approach between contacting pinion and gear teeth,i.e., the composite contact deformation in Point D under load

acting in the same point. ( ) ( )DF zp,zp are the tooth loads acting

in Positions F and D, respectively.As the contact points are at different distances from the pin-

ion/gear axis, the transmitted torque is defined by the equation

( ) dzzrTtt

t it

Ni

1i L

F0FF ==

=

tpr

r

(17)

where Fr is the distance of the loaded Point F to the gear axis,

F0tr

is the tangent unit vector to the circle of radius Fr , passing

through the loaded Point F in the transverse plane of the gear,

and tN is the number of gear tooth pairs instantaneously in

contact.

The load distribution on each line of contact can be calcu-lated by solving the nonlinear system of Eqs. (14)-(17). Anapproximate and iterative technique is used to attain the solu-tion. The contact lines are discretized into a suitable number ofsmall segments, and the tooth contact pressure, acting along a

segment, is approximated by a concentrated load, Fr

, acting inthe midpoint of the segment. The actual load distribution, de-

fined by the values of loads Fr

is obtained by using the suc-cessive-over-relaxation method. In every iteration cycle, asearch for the points of the "potential" contact lines that could

be in instantaneous contact is performed. For these points, thefollowing condition should be satisfied:

( )( )

( )

( )

( )zt

t

zt

i,iD

0

k

i22

i,in

r

y

r

earr

rrr

(18)

where ti is the identification number of the contacting tooth

pair, zi of the segment.

The details of the method for load distribution calculationin spiral bevel gears are described in Ref. [23].

RESULTS

A computer program, based on the theoretical backgroundpresented, has been developed. By using this program, theinfluence of the character and order of polynomial functions (9)and (10) conducting the variations of the cradle radial setting

pe and the modified roll for pinion generation on transmission

errors, motion graphs and maximum tooth contact pressure wasinvestigated.

The calculation was carried out for the spiral bevel gear pairof design data given in Table 1. The basic machine-tool setting

parameters for finishing the pinion and the gear teeth blankswere calculated by the method used in Gleason Works and bythe method presented by Argyris et al. [2], and are given inTables 2 and 3.

Table 1 Pinion and gear design data

Pinion Gear

Number of teeth 13 50Module, mm 5Outside diameter, mm 76.746 251.224Face width, mm 30Pitch apex to crown, mm 123.473 30.146Mean spiral angle, deg 35Pitch angle, deg 14.5742 75.4258Face angle of blank, deg 17.7067 76.9478Root angle, deg 13.0522 72.2933Addendum, mm 6.068 2.432Dedendum, mm 3.432 7.068

Working depth, mm 8.500Whole depth, mm 9.500

Table 2 Pinion machine-tool settings

Concave Convex

Point radius of the cutter, mm 86.117 93.433Cutter blade angle, deg 18.5 21.5Machine root angle, deg 13.0522 13.0522Basic cradle angle, deg 51.4895 47.2527Sliding base setting, mm 0.5602 -0.3882Machine center to back, mm -2.4805 1.7189Basic radial, mm 92.8531 99.3590Blank offset, mm 0.2001 -0.7347Ratio of roll 3.8470 4.0560

Table 3Gear machine-tool settings

Cutter diameter, mm 180Cutter point width, mm 2.79Cutter blade angle, deg 20Machine root angle, deg 72.2933Basic cradle angle, deg -49.7814Basic radial, mm 96.5017Ratio of roll 1.0317

Two cases were investigated: in the first case, the coeffi-cients in Eqs. (9) and (10) were constant throughout the wholegeneration process of one pinion tooth-surface, in the secondcase, the coefficients were different for the generation of the

pinion tooth-surface on the two sides of the initial contact point.In Fig. 3 the motion graphs are presented for the case, when

the cradle radial setting pe variation is conducted by the same

polynomial functions up to third order throughout the wholegeneration process of a pinion tooth flank:

( ) ( ) ( )30cpcp2

0cpcp0cpcp0pp 04.0013.0094.0ee ++= (19)

The maximum angular displacements of the driven gearmember for different orders of the polynomial function are: in

the case of 1st , 2nd and 3rdorders sec,arc231.2max2 = 2.206

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arcsec, 2.206 arcsec, respectively. For the basic cradle radialsetting (given in Table 3), the maximum transmission error is7.039 arc sec. Therefore, a second order polynomial function issatisfactory, while the maximum transmission error is reduced

by 68 %.

In the next trial, the cradle radial setting variation was con-ducted with different polynomial functions up to 5thorder forthe generation of the pinion tooth-surface on the two sides ofthe initial contact point. The obtained motion graphs are shownin Fig. 4. The optimized polynomial functions were:

- for the generation of the part of pinion tooth flank between the toe of the tooth and the initial contact point:

( ) ( ) ( ) ( ) ( ) ( )50cpcp4

0cpcp

3

0cpcp

2

0cpcp0cpcp0p

1

p 6925.64.6025.0ee ++= (20)

- for the generation of the part of pinion tooth flank between the initial contact point and the heel of the tooth:

( ) ( ) ( ) ( ) ( ) ( )50cpcp4

0cpcp

3

0cpcp

2

0cpcp0cpcp0p

2

p 2475.8275.0125.0133.0ee +++= (21)

0

1

2

3

4

5

6

7

8

0 0,2 0,4 0,6 0,8 1

N 1/2

2

[arc

sec]

Basic

Linear

Second-order

Third-order

Fig. 3:Motion graphs for the case when the variation of theradial setting is conducted by the same polynomial functions

up to third order throughout the whole generation process

Table 4 The transmission errors for different orders of thepolynomial function applied for radial setting variation

Order of thepolynomial

function

Maximumtransmission

error [arc sec]

Basic 7.039

1st

order 3.4582

ndorder 1.868

3rdorder 1.460

4thorder 1.396

5thorder 1.358

The maximum angular displacements of the driven gearmember, for different orders of the polynomial function aregiven in Table 4. It can be concluded that by the 5 thorder poly-nomial functions (20) and (21), the maximum transmissionerror can be reduced by 81 %. The variation of the cradle radial

setting pe for the generation of pinion tooth, conducted by the

two 5thorder polynomial functions, is presented in Fig. 5.

0

1

2

3

4

5

6

7

8

0 0,2 0,4 0,6 0,8 1

N 1/2

2[arcsec]

Basic

Linear

Second-order

Third-order

Fourth-order

Fif th-order

Fig. 4:Motion graphs for the case when the variation of the

radial setting is conducted by two different polynomialfunctions on the two sides of the initial contact point

Table 5 Coefficients of the polynomial functions and thetransmission errors

N1 = 9 N1 = 19

Betweenthe toeand the

initialcontactpoint

Betweenthe

initial

contactpointand the

heel

Betweenthe toeand the

initialcontactpoint

Betweenthe initialcontact

point andthe heel

ce1 0.088 0.197 0.003 0.075

ce2 -0.40 -0.64 -4.6 -0.006

ce3 0 0.50 -19 -4.48

ce4 33 33.2 -2 12

ce5 -2 18 0 -1

max2 : ep=const. 8.742 arcsec 3.672 arcsec

max2 : epvarried 2.130 arcsec 1.332 arcsec

Reduction 76 % 64 %

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eration is conducted by the same polynomial functions up to 3rd

order throughout the whole generating process:

( ) ( ) ( )30cpcp2

0cpcp0cpcpgp1 0002.00276.0i +=

(22)

In the second case, the optimal variation of the modified rollfor pinion tooth flank generation is conducted by two different

polynomial functions up to 4th order on the two sides of theinitial contact point:

( ) ( ) ( ) ( ) ( )40cpcp3

0cpcp

2

0cpcp0cpcpgp

1

1 0075.0065.02425.0i += (23)

( ) ( ) ( ) ( ) ( )40cpcp3

0cpcp

2

0cpcp0cpcpgp

2

1 0075.00205.001925.0i = (24)

0

1

2

3

4

5

6

7

8

0 0,2 0,4 0,6 0,8 1

N 1/2

2[arcsec]

Basic

Second-order

Third-order

Fourth-order

Fig. 9:Motion graphs for the case when the variation of themodified roll for pinion tooth flank generation is conducted

by two different polynomial functions on the two sidesof the initial contact point

Table 6 The transmission errors for different orders of thepolynomial function applied for modified roll variation

Maximum transmission error

max2 [arcsec]Order of thepolynomial

functionI. Case II. Case

1storder 7.039 7.039

2nd

order 5.580 3.059

3rd

order 5.542 2.139

4thorder 2.138

Reduction 21% 61%

The corresponding maximum angular displacements of thedriven gear member are given in Table 6. Again, the use ofdifferent polynomial functions on the two sides of the initialcontact point performs better results: the reduction of the maxi-

maximum transmission error is 61% against 21% when thesame polynomial function is used throughout the whole genera-tion process. The variation of the modified roll for pinion tooth-surface generation, conducted by the two 4 thorder polynomialfunctions, is presented in Fig. 10.

3,81

3,82

3,83

3,84

3,85

3,86

40 45 50 55 60

[deg.]

igp

Basic

Fourth-order

Fig. 10:Variation of the modified roll for the generation of onepinion tooth flank conducted by two different 4thorder poly-nomial functions on the two sides of the initial contact point.

The influence of misalignments, as are (Fig. 1): the pinionoffset ( a ), the displacements of the pinion along the pinion

axis ( b ) and along the gear axis ( c ), and the angular mis-alignments of the pinion axis in the horizontal ( h ) and in the

vertical plane ( v ), on transmission errors of spiral bevel gearswith pinion whose teeth are processed by the variation of cradle

radial setting ( pe ) and of modified roll ( gpi ) is investigated and

the obtained results are presented in Figs. 11-15.It can be noted that in most cases, the modifications intro-

duced into the pinion tooth-surface by the variation of the cra-dle radial setting, yield to higher transmission error reductions.In Fig. 15 it can be seen that the transmission errors are insensi-tive to angular misalignments of the pinion axis in the vertical

plane.

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0

1

2

3

4

5

6

7

8

9

10

-0,1 -0,05 0 0,05 0,1

a [mm]

2max

[arc

sec

]

No corrections

Corrections by "ep"

Corrections by "igp"

Fig. 11:Influence of pinion offset on transmission errors

0

5

10

15

20

-0,1 -0,05 0 0,05 0,1

b [mm]

2max

[a

rcsec]

No corrections

Corrections by "ep"


Fig. 12:Influence of displacements of the pinion along its axison transmission errors

The influence of the transmitted torque on transmission er-rors is investigated, also (Figs. 1619). The modifications,introduced by the variation of the cradle radial setting, have astronger effect on the reduction of transmission errors, espe-cially in the case of smaller transmitted torques (Fig. 16). InFig. 17, the motion graphs for the spiral bevel gear pair with a

pinion finished by the basic machine-tool setting, loaded withdifferent torques, is presented. It can be noted that the angulardisplacements of the driven gear are much bigger for moderatetorque values. By the use of tooth modifications, introduced by

the variation of cradle radial setting conducted by differentoptimal polynomial functions on the two sides of the initialcontact point, the transmission errors can be considerably re-duced, especially in the case of moderate torque values (Fig.18). When the modified roll variation is applied, the angulardisplacements of the driven gear significantly increase in thecase of light loads (Fig. 19).

0

2

4

6

8

10

12

-0,1 -0,05 0 0,05 0,1

c[mm]

2max

[arcs

ec]

No corrections

Corrections by "ep"


Fig. 13:Influence of displacements of the pinion along the gearaxis on transmission errors

0

1

23

4

5

6

7

8

9

10

11

-0,1 -0,05 0 0,05 0,1

h[deg.]

2max

[arcsec]

No corrections

Corrections by "ep"


Fig. 14:Influence of angular misalignment of the pinion axis in

the horizontal plane on transmission errors

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1

2

3

4

5

6

7

8

-1 -0,5 0 0,5 1

v[deg.]

2max

[arcs

ec]

No corrections

Corrections by "ep"


Fig. 15:Influence of angular misalignment of the pinion axis in

the vertical plane on transmission errors

0

1

2

3

4

5

6

7

8

0 50 100 150 200

T [Nm]

2m

ax

[arcsec] No corrections

Corrections by "ep"


Fig. 16:Influence of transmitted torque on transmission errors

The maximum tooth contact pressure can be reduced onlymoderately by the use of machine-tool setting variations. Incomparison with the spiral bevel gear pair, whose pinion isgenerated by the use of the basic machine-tool setting (Fig. 20),the maximum tooth contact pressure can be reduced by 4% inthe case of optimal cradle radial setting variation based on thereduced transmission errors (Fig. 21), and by 7% when thevelocity ratio in the kinematic scheme of the machine-tool forthe generation of pinion tooth-surface is optimally varied (Fig.22).

0

1

2

3

4

5

6

7

8

0 0,25 0,5 0,75 1

N1/2

2[arcs

ec]

T= 5 Nm

T= 10 Nm

T= 25 Nm

T= 50 Nm

T= 80 Nm

T= 120 Nm

T= 200 Nm

Fig. 17:Influence of transmitted torque on motion graphs forthe case when no modifications are introduced into

the pinion tooth-surface

0

1

2

3

4

5

0 0,25 0,5 0,75 1

N

1

/2

2[arcsec]

T=5 Nm

T=10 Nm

T=25 Nm

T=50 Nm

T=80 Nm

T=120 Nm

T=200 Nm

Fig. 18:Influence of transmitted torque on motion graphs when

the optimal cradle radial setting variation is appliedfor pinion teeth finishing

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0

2

4

6

8

10

12

0 0,25 0,5 0,75 1

N 1/2

2[arcsec]

T=5 Nm

T=10 Nm

T=25 Nm

T=50 Nm

T=80 Nm

T=120 Nm

T=200 Nm

Fig. 19:Influence of transmitted torque on motion graphs whenoptimal modified roll variation is applied for pinion teeth fin-

ishing

Fig. 20:Tooth contact pressure distribution when the pinionteeth are generated with the basic machine-tool setting

CONCLUSIONS

A method for the determination of the optimal polynomialfunctions for the conduction of machine-tool setting variationsin pinion teeth finishing in order to reduce the transmissionerrors in spiral bevel gears is presented. Polynomial functionsof order up to five are applied to conduct the variation of thecradle radial setting and of the cutting ratio in the process for

pinion teeth generation. On the basis of the obtained results thefollowing conclusions can be made.

1. The transmission errors up to 81% can be reduced by theuse of the optimal variation of the cradle radial setting in piniontooth processing, conducted by two different polynomial func-tions of 5thorder on the two sides of the initial contact point.

2. The transmission errors can be also considerably reducedby the use of the optimal variation of the cradle radial setting inthe case of misalignments inherent in the spiral bevel gear pair,and for different load levels.

3. The maximum tooth contact pressure can be reducedonly moderately: in the case of the optimal variation of themodified roll, the reduction is 7%, and in the case of the varia-

tion of the cradle radial setting 4%.4. The investigations have shown that by the combined

variation of the cradle radial setting and of the modified roll, nofurther reduction of transmission errors can be achieved.

Fig. 21:Tooth contact pressure distribution when the pinionteeth are generated with optimal cradle radial setting variation

Fig. 22:Tooth contact pressure distribution when the pinionteeth are generated with optimal modified roll variation

ACKNOWLEDGMENTS

The author would like to thank the Hungarian Scientific Re-search Fund (OTKA) for their financial support of the researchunder Contract No. T 035207.

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